Summary
The Beta distribution in n-dimensions is introduced to describe the
proportions of the mineralogical components existing in a certain stratigraphic
interval (the porosity is included as a "mineralogical component"). The
justification for doing so is empirical. The model allows the calculation of
well-logging parameters, such as GRma, GRsh, and shale
density, without having to introduce them by "eye." It also allows the
probabilistic calculation of the rock composition at each depth when there are
more mineralogical components than logs: that is, there is a shortage of
equations. In addition to this, the Beta model can be used to test the
hypothesis that the relationship between any two components can be regarded as
random, which should have applications in reservoir characterization.
Introduction
Sedimentary rocks may be described ultimately as a mixture of minerals and
pores. For a given lithological column, it is possible using well logs to
calculate the composition of the rocks at discrete points. We may ask which
should be the probability distribution of the volume fraction of each mineral
component (with the porosity included as a "mineral component") along this
lithological column. This distribution should satisfy at least the following
conditions:
(a) The values of each of the components
should range between 0 and 1.
(b) The sum of all the components should be
equal to 1, for all points.
The well-known Beta distribution, which is also known as the Dirichlet
distribution in the multidimensional case (Gelman et al. 2003), satisfies these
requirements. Although, in theory, this distribution allows for a porosity of
1, in practice the values of the parameters of the distribution are such that
very high porosities are extremely unlikely. There are also empirical
observations that support the use of this distribution to model rocks. It is
quite frequent to see histograms of the gamma ray (GR) log across more or less
"homogeneous" intervals, which are clearly unimodal and asymmetrical (i.e.,
they are skewed). If we assume that the GR log is sensitive to only one
component (the "shale"), then, if the shale volume fraction is
Beta-distributed, the character of the GR log can be explained easily.
In summary, despite the lack of a sound theoretical background, there are
some numerical characteristics and empirical observations that justify the
introduction of this distribution.
© 2009. Society of Petroleum Engineers
View full textPDF
(
1,226 KB
)
History
- Original manuscript received:
31 January 2007
- Meeting paper published:
15 April 2007
- Revised manuscript received:
23 February 2009
- Manuscript approved:
30 March 2009
- Published online:
29 September 2009
- Version of record:
31 December 2009
View Cart
